72 CHAPTER 7

In other words,

•

a

p

=−1ifS(F

p

) contains no elements.

•

a

p

= 0ifS(F

p

) contains one element. This happens only

if a is a multiple of p.

•

a

p

= 1ifS(F

p

) contains two elements.

•

Because

a

p

depends only on a modulo p,

a

p

=

a + kp

p

for any integer k.

4

You might wonder why Legendre chose to shift by 1, rather than

just letting

a

p

be the number of elements in S(F

p

). One answer is

the following easy pair of equations:

0

p

= 0,

1

p

= 1.

But there is more to it than that.

Digression: Notation Guides Thinking

There are some very famous instances of how proper choices of

notation can help speed the progress of mathematics. Someone

outside of the ﬁeld might not even realize that there is a distinction

between the notations that mathematicians use and the subject

itself.

For example, algebraic notation—x, y, and all the rest—is

notation, not mathematics. Greek mathematicians did not have

that symbolism, and they were able to express problems in words

and solve them without using symbols. Today, of course, the use of

x to stand for an unknown thing is so widespread that it has passed

into popular culture. Using letters to stand for quantities makes it

4

Pay careful attention to this fact; we will be using it repeatedly, both in proofs and

computations.

Start Free Trial

No credit card required